Rubber or latex gloves are being relied on more and more as protection against contamination and disease, especially with the recent concern over AIDS. Rubber gloves are used in such great quantities that their packaging is often made to double as a dispenser. Some such dispensers use simply a hole in the top of the container. Others have the hole covered by flexible sheets that abut each other along a parting line to establish a slot through which the gloves can be easily withdrawn. One problem with such devices is that they permit ozone, which is present in many environments where rubber gloves are used, to enter the box and attack the rubber gloves. Within a short period of time after exposure to ozone rubber gloves become brittle, deteriorate and are easily torn. One approach to preventing ozone from entering the container and deteriorating the rubber gloves uses a pair of overlapping flaps which cover the dispenser hole. Usually these flaps are glued to the underside of the top which bears the hole. In that construction, however, these flaps are easily dislodged or torn from their attachment to the container. This can occur when the user inserts a hand too far into the hole, thereby spreading the flaps apart excessively, or when the user grabs in such a way as to pull on one flap laterally to open the hole wide to allow the other hand to easily enter the hole and grab a glove.